Accurate characterization and simulation of hydrocarbon reservoirs can assist in maximizing the success of finding, developing, and producing hydrocarbons. Sedimentary process simulation which is based on physics and hydrodynamics is an advanced tool used to simulate reservoir deposits. One of the most challenging tasks for sedimentary process simulation is to find the appropriate input parameters, including the initial and boundary conditions, that allow the simulation model to generate simulated reservoir deposits that are consistent with the measured/observed reservoir data. Unfortunately, these input parameters are generally not measurable, not collectable, and even not retrievable because they disappeared millions of years ago during reservoir formation. Typically, the only media where data can be measured and collected for the simulation model are the reservoir deposits, the product of the sedimentary processes. Deposit data are generally collected through seismic surveys, (exploration, development, and production) well logs, and other means. Because seismic and well data are the measures of model output (response) and cannot be used directly as model input, they are used to infer a set of appropriate input parameters for sedimentary process simulation. The inference process is called conditioning.
Conditioning sedimentary process simulation to seismic and well data is a type of inverse problem. Inverse problems have been studied in science and engineering for decades. They are generally difficult to solve because they are commonly ill-posed (i.e., (1) the solution may not exist, (2) the solution may not be unique, and (3) the solution may not depend continuously on the data). Solving an inverse problem in the earth sciences, especially in sedimentary process simulation, is even more difficult because the data required to constrain the problem is extremely scarce. Conditioning sedimentary process simulation to seismic and well data is considered by some researchers to be very difficult, unsolved, and even impossible (Burton et al., 1987).
There are no effective or robust methods for conditioning sedimentary process simulation to seismic and well data today. Traditionally, an inverse problem in science and engineering is formulated as an optimization problem and then solved using a variety of optimization methods, e.g., a gradient-based or direct search method. Some of these optimization methods have been applied to conditioning sedimentary process models. However, they have not yet demonstrated much success. One of the reasons that hinder the success of conditioning is that optimization-based methods do not directly address the ill-posed nature of conditioning sedimentary process simulation.
Because it is very difficult to directly solve a nonlinear numerical model inversely, a conditioning (inverse) problem is traditionally formulated in the form of an objective (fitness) function and the problem is solved by using algorithms developed in the field of optimization. The objective function measures the misfit between the model responses and observations, and the disagreement between known knowledge and the model. It is formulated as followsJ(y,yo,μ(x),μo,x)=f1(∥y−yo∥)+f2(∥μ(x)−μo∥)  (1)where J is the objective function that is a function of y, yo, μ(x), μo, and x; y is a response(s) of the model; yo is an observation(s) of the response(s) of the physical systems to be modeled; μ(x) is a measure(s) of the model parameters; μo is a known measure(s) of the model parameters for the physical systems; x represents an input parameter(s); ∥ ∥ is a norm that measures the length, size, or extent of an object; f1, f2 are some given functions. Most conditioning (inverse) problems use the misfit between the model responses and observations (the first term of the right hand side of Equation (1)) only. However, to honor the known knowledge about the physical systems to be modeled, e.g. in the Bayesian approach, the second term of the right hand side of Equation 1 is needed. Under the Bayesian approach, known knowledge about physical systems is called the a priori information about the systems. Equation 1 is solved by using an optimization algorithm to find the solution where the value of the objective function is the smallest. Most of the current conditioning methods in geology are optimization-based.
Conditioning numerical models in geology is considered to be difficult, unsolved, and even impossible. An extensive review on four decades of inverse problems in hydrogeology can be found in Marsily et al.'s article (2000). After analysis of the inversion of eustasy from the stratigraphic record, Burton et al. (1987) concluded that conditioning (inversion) of stratigraphic data is impossible. The major reasons that prevent the successful application of inversion are non-uniqueness and non-distinguishableness. Marsily and Burton studied the effect of subsidence, sediment accumulation, and eustasy on sequence stratigraphy and found that different combinations of the three parameters result in the same stratigraphic record, i.e., the non-uniqueness of the inverse problem. They also found that it is difficult to distinguish the effect of one parameter on the stratigraphy from the others. At best, all they could deduce is the sum of the three parameters rather than the individual parameters.
Some researchers (e.g., Cross et al., 1999 and Karssenberg et al., 2001) believed that conditioning numerical models to stratigraphic data is possible. Cross et al. developed an inverse method using a combination of linear and nonlinear solutions developed by Lerche (1996) to solve the conditioning problem of stratigraphic simulation. They demonstrated their method and some of their results with two dimensional deposits in 2001 in U.S. Pat. No. 6,246,963 B1. They later in 2004 extended their method from 2D (two dimensions) to 3D (three dimensions) in U.S. Pat. No. 6,754,558 B2. Even though Cross et al.'s method can avoid being trapped at local minima which is the common problem of gradient-based optimization methods, the method cannot address the non-uniqueness problem of conditioning process-based models.
Lazaar and Guerillot's United States Patent (2001), U.S. Pat. No. 6,205,402 B1, presented a method for conditioning the stratigraphic structure of an underground zone with observed or measured data. They used a quasi-Newtonian algorithm (a gradient-based method) to adjust the model parameters until the difference between model results and observed/measured data is minimized. Unfortunately, like most gradient-based methods, Lazaar and Guerillot's method suffers the problem of convergence at local minima. Similar to Cross et al.'s method, Lazaar and Guerillot's method cannot address the common non-uniqueness issue in conditioning of process-based models.
Karssenberg et al. (2001) proposed a trial-and-error method for conditioning three dimensional process-based models (simplified mathematic models without full physics) to well data. Because their goal was to show the possibility of conditioning process-based models to observational data in principle rather than to demonstrate the effectiveness and efficiency of the method for real-world problems, their method, as they stated, was computationally intensive and not ready for real-world problems.
Bornholdt et al. (1999) demonstrated a method to solve inverse stratigraphic problems using genetic algorithms for two simple (two-dimensional cross section) models. Although their method is very time-consuming (exploring 1021 different stratigraphic models), the authors claimed that the method will eventually be attractive considering the exponential decay of computing costs and the constant increase of manpower costs. Their method is not ready for real world applications due to problems of efficiency and effectiveness using current computing technology.
Genetic algorithm (GA) is a powerful direct search based optimization method originally inspired by biological evolution involving selection, reproduction, mutation, and crossover (Golden and Edward, 2002). Because it searches for global minima and does not require calculation of the derivatives of the objective function (Equation 1), the method has attracted a broad attention from many different industry and academic fields, such as geology (Wijns et al., 2003), geophysics (Stoffa and Sen, 1991; Sambridge and Drijkoningen, 1992), hydrology (Gentry et al., 2003; Nicklow et al., 2003), and reservoir engineering (Floris et al., 2001; Schulze-Riegert et al., 2002; and Schulze-Riegert et al., 2003). GA is the primary conditioning tool used in sequence stratigraphic modeling. The main advantage of GA is that it finds the global minima while addressing the non-uniqueness property of the solutions with some degree of uncertainty. The major complaints of GA are its time-consuming nature and its convergence problems.
Another direct search based optimization method is the neighborhood algorithm (NA). Unlike GA, NA makes use of geometrical constructs (Voronoi cells) to derive the search in the parameter space (Sambridge, 1999a and 1999b). Recently, Imhof and Sharma (2006 and 2007) applied NA to seismostratigraphic inversion. They demonstrated their method using a small and simple diffusion model.
An attractive gradient-based method for conditioning is the adjoint method (Schlitzer, 2000) that solves the forward equations and the backward adjoint equations for the inverse problem. The major advantage of adjoint method over other methods is that it is independent of the number of input parameters to be inverted. As a result, the method can be used for very large parameter identification problems. Adjoint methods originally from optimal control theory have been used in reservoir engineering in the last thirty years (Chen et al., 1974; Chavent, 1975; Li et al., 2003; Sarma et al., 2005; and Vakili et al., 2005). Adjoint methods have also been applied in hydrogeology (Neuman and Carrera, 1985; Yeh and Sun, 1990) and hydraulic engineering (Piasecki and Katopades, 1999; Sanders and Katopodes, 2000). However, adjoint methods have not been used in conditioning of sedimentary process simulation. Like other gradient-based methods, adjoint methods have the danger of being trapped at local minima.
The ensemble Kalman filter (EnKF) is a data assimilation method that has been applied in atmospheric modeling (Houtekamar and Mitchell, 1998; Kalnay, 2003, pp. 180-181), oceanography (Evensen, 2003), and reservoir engineering (Gu and Oliver, 2005; Nœvdal et al., 2005; Zafari and Reynolds, 2005; Lorentzen et al., 2005; and Zhang et al., 2005). The major advantages of EnKF are the dynamic update of models and the estimation of model uncertainties. There are some limitations for EnKF (Kalnay, 2003, page 156): if the observation and/or forecast error covariances are poorly known, if there are biases, or if the observation and forecast errors are correlated, the analysis precision of EnKF can be poor. Other complaints include sensitivity to the initial ensemble, destroying geologic relationships, non-physical results, and the lack of global optimization etc. Vrugt et al. (2005) proposed a hybrid method that combines the strengths of EnKF for data assimilation and Markov Chain Monte Carlo (MCMC) for global optimization. Because EnKF requires dense observations, both in space and in time, which may be generally unavailable in geological processes, its application in sedimentary process simulation is still an open question.
Although optimization-based conditioning methods have been successfully applied in many fields of science and engineering, their success in the conditioning of the sedimentary process simulation is limited. One of the reasons is that the conditioning problem in the sedimentary process simulation is ill-posed and the success of a conditioning method depends on its ability to address the ill-posed problem. Some of optimization-based conditioning methods, e.g., GA and NA, have attacked the problem of non-uniqueness (one of the issues of ill-posedness) but the other issues (the existence of solution and the continuity of the parameter space) of the ill-posed problem are not addressed. If the three ill-posedness issues are not fully addressed, conditioning may not give a meaningful solution. As a result, the solution obtained may have no predictive power. An example of this issue was reported by Tavassoli et al. (2004).
Tavassoli et al. (2004) used a very simple reservoir model to demonstrate a fundamental problem in history matching (conditioning reservoir models to historical production data). The reservoir model used is a layer-cake 2D cross-section model with six alternating layers of good- and poor-permeabilities and a vertical fault cutting through the middle of the model. Three model parameters (fault throw h, good permeability kg, and poor permeability kp) are chosen to test the history matching problem. The parameter space is a cube with 0≦h≦60, 100≦kg≦200, and 0≦kp≦50. The “truth” model is (h, kg, kp)=(10.4, 131.6, 1.3). Through their search, they found that the best history-matched model (i.e., the model with the smallest value of the objective function) is (h, kg, kp)=(33.1, 135.9, 2.62). It is apparent that the best history-matched model has a poor estimation for the “truth” model. As confirmed by their simulation, the best history-matched model gives a poor prediction. They conclude that the best match of a model to observation data does not necessarily result in the best model for prediction.